EVENT DETAILS AND ABSTRACT

Title: Extension problem for positive-definite generalized Toeplitz kernels
Speaker: Professor Miron Bekker
Speaker Info: Northwestern University
Brief Description:
Special Note:
Abstract:

Positive-definite generalized Toeplitz kernels (GTK) appear in a natural way in some problems of harmonic analysis ( the Hilbert transform and the theorem of Helson and Szego), in scattering theory and in the theory of generalized stochastic processes. We prove that for a matrix valued kernel to be positive, it is necessary and sufficient that it admit a Bochner-type integral representation. From this it follows that that any positive definite GTK initially defined on an interval or the positive semi-axis extends to the real line while preserving its structure and positive-definiteness, generalizing a theorem of Bochner and Krein. We also discuss conditions for the uniqueness of extensions and descritpion of all possible extensions when the uniqueness conditions are not met. The methods involve the theory of self-adjoint extensions of symmetric operators and the technique of rigged Hilbert spaces.